DESCRIPTIVE GEOMETRY. 35 
two points in any straight line are known, the line itself will be deter- 
" nined. | | 
We can imagine the horizontal plane to be revolved about its axis in such 
a manner as to form an angle (= 2R) with the other plane, leaving only 
one plane upon which objects are to be projected. If a line out of this 
plane is to be projected upon it, perpendiculars must be let fall from the two 
extremities upon the horizontal plane; the straight line connecting the feet 
of these perpendiculars is called the projection of the line. 
As a straight line is determined by two of its points, and as curved lines 
require several, every curved line may be considered as consisting of 
infinitely small straight lines. The projection of a curved line, then, is the 
same as that of a straight line, only requirmg more points in the line. Thus 
in pl. 4, fig. 23, let 1234 9 indicate the position of a line in horizontal 
projection—we here suppose the horizontal plane to be revolved—and I’ 2: 
3! 9’ be the position of the line with respect to the vertical plane zy; 
the projection of the line is now to be found. First of all a number of points 
is to be assumed in the line 1- 9, the same determined in the line 1’ 9’, 
‘and perpendiculars drawn to the corresponding plane, which determine the 
feet. Prolong these perpendiculars beyond their feet and they will intersect 
each other, the points of intersection of the corresponding perpendiculars 
forming the corresponding points of the projection. Thus, from the inter- 
section of the perpendiculars'3 and 3’, the point 3° lying in the projection is 
‘ascertained. When all the points are found, the projection will be obtained 
by joining 1?, 2?, 3? 9°, this will be the line. 
As surfaces are bounded by lines, we can obtain the projection of surfaces 
by finding the lines inclosing them.. In fig. 21 let abcd be the position gf 
a surface in plan, a’ c* is the position of ac in elevation. To determine the 
two corners b* and d’ on this line, project b and d upon ac, in b’ and d?, and 
take off these points on a’ c*.. Drawing perpendiculars from the four 
extremities of the two horizontal figures, we shall have the points a’, b’, c’, d’ 
-as the corners of the projection, which is itself obtained by connecting the 
corners by straight lines. 
If the figure be bounded by curved lines, a mode of proceeding similar to 
that.employed in the case of straight lines will be necessary. In fig. 22 let 
ab be the view of a circular plane, in ground plan, f° f*, the same in eleva- 
‘tion. It is well known that the end view of a circle perpendicular to a 
plane appears as a straight line, this in the ground plan being the horizon- 
tal, and in the elevation the vertical diameter. We must thus, first of all, 
find the points in the curved line which are to be projected. For this pur- 
pose describe the two semicircles, divide them into an equal number of 
equal parts, for instance, inc’, d’, e’, &c., and in a‘, c’, d*, &c., and project 
these upon the diameter ; we shall thus obtain the points a, c,d, &c., and 
a’, c’, d’, &c. By drawing the lines of projection from the like named 
points, we shall obtain the projected points of half the curved line. Thus. 
for instance, from the lines of projection from d and d’, we get the point d’, 
and as d and h lie at an equal distance from the centre, we obtain by means 
of the lines from A and a’ the point h* symmetrical with d*. After the 
35 
